Bounds and E-values for Marginal Causal Effects

Unmeasured confounding is an important obstacle when estimating causal effects from observational data. Ding and VanderWeele (EPIDEMIOLOGY 2016;27:368) derived bounds for causal effects, based on sensitivity parameters that quantify the maximal strength of unmeasured confounding. These bounds translate to the popular E-value metric, which quantifies the magnitude of unmeasured confounding required to “explain away” an observed association. While Ding and VanderWeele mainly focused on conditional (on measured confounders) causal effects, they also outlined how their method might be used for marginal causal effects. However, this requires specification of the sensitivity parameters at each level of the measured confounders, which is impractical in high-dimensional settings, and it yields overly conservative bounds that lack a natural E-value analog. In this article, we propose novel bounds for marginal causal effects based on Ding and VanderWeele’s sensitivity parameters. The proposed bounds only require the analyst to specify the maximal values of the sensitivity parameters across all levels of the measured confounders, thus substantially reducing dimensionality. Furthermore, the proposed bounds are often narrower than Ding and VanderWeele’s bounds, and they translate naturally into an E-value for marginal causation. We show how the proposed bounds can be estimated using standard regression techniques, and we illustrate through an application to publicly available data, with accompanying R code provided.